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Tag Archives: semisimple rings
Polynomials and Representations XXXV
Schur-Weyl Duality Throughout the article, we denote for convenience. So far we have seen: the Frobenius map gives a correspondence between symmetric polynomials in of degree d and representations of ; there is a correspondence between symmetric polynomials in and polynomial … Continue reading
Modular Representation Theory (IV)
Continuing our discussion of modular representation theory, we will now discuss block theory. Previously, we saw that in any ring R, there is at most one way to write where is a set of orthogonal and centrally primitive idempotents. If such an … Continue reading
Modular Representation Theory (I)
Let K be a field and G a finite group. We know that when char(K) does not divide |G|, the group algebra K[G] is semisimple. Conversely we have: Proposition. If char(K) divides |G|, then K[G] is not semisimple. Proof Let , a two-sided … Continue reading
Projective Modules and Artinian Rings
Projective Modules Recall that Hom(M, -) is left-exact: for any module M and exact , we get an exact sequence Definition. A module M is projective if Hom(M, -) is exact, i.e. if for any surjective N→N”, the resulting HomR(M, N) → HomR(M, N”) is … Continue reading
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Tagged artinian, free modules, left-exact, projective modules, semisimple rings, splitting lemma
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Jacabson Radical
Recall that the radical of the base ring R is called its Jacobson radical and denoted by J(R); this is a two-sided ideal of R. Earlier, we had proven that a ring R is semisimple if and only if it is artinian and J(R) = … Continue reading
Posted in Notes
Tagged artinian, hopkins-levitzki, jacobson radical, matrix rings, nilpotent ideals, noetherian, semisimple rings
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Radical of Module
As mentioned in the previous article, we will now describe the “bad elements” in a ring R which stops it from being semisimple. Consider the following ring: Since R is finite-dimensional over the reals, it is both artinian and noetherian. However, R is not … Continue reading
Posted in Notes
Tagged algebra, artinian, jacobson radical, matrix rings, modules, radical of modules, semisimple rings
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Noetherian and Artinian Rings and Modules
We saw the case of the semisimple ring R, which is a (direct) sum of its simple left ideals. Such a ring turned out to be nothing more than a finite product of matrix algebras. One asks if there is a … Continue reading
Posted in Notes
Tagged algebra, artinian, noetherian, semisimple rings, simple modules
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The Group Algebra (III)
As alluded to at the end of the previous article, we shall consider the case where K is algebraically closed, i.e. every polynomial with coefficients in K factors as a product of linear polynomials. E.g. K = C is a common choice. Having assumed … Continue reading
Posted in Notes
Tagged character theory, division rings, group algebras, quaternions, semisimple rings, simple modules
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The Group Algebra (I)
[ Note: the contents of this article overlap with a previous series on character theory. ] Let K be a field and G a finite group. The group algebra K[G] is defined to be a vector space over K with basis , where “g” here is … Continue reading
Posted in Notes
Tagged character theory, group actions, group algebras, modules, representation theory, semisimple rings, simple modules
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Structure of Semisimple Rings
It turns out there is a nice classification for semisimple rings. Theorem. Any semisimple ring R is a finite product: where each is a division ring and is the ring of n × n matrices with entries in D. Furthermore, the … Continue reading
Mathematics and Such 